Quiz Question

One of a continuing series

Here’s something that came up a few years ago. It has to do with GPS technology, but you don’t need to be GPS-savvy to appreciate it. It goes like this.

GPS employs 32 satellites, whose position is at all times precisely known. Each satellite broadcasts to Earth continuously. The signal contains a lot of stuff, but the critical information from each satellite is:

• I am here.
• The time is …

A GPS navigation receiver doesn’t need to know the direction the signal is coming from. All that is necessary to determine your position is the preceding information from each of three or four (four is best) satellites.

From the information received, a navigator can determine where all satellites were at the same time. And it can determine how long it took each satellite’s signal to reach the receiver. Knowing the speed of propagation of the radio signal, the receiver can compute the distance to each of three (or four) known points in space and therefore compute its own position in 3-D space. If it knows the speed of propagation of the radio signal.

The problem is the speed of propagation through the atmosphere to the receiver is not constant. It varies due to the presence of free electrons in the atmosphere. There are two solutions to this difficulty. One is to incorporate an atmospheric model into the receiver’s computation, and this is done. It’s called the Klobuchar model, after the person who developed the mode. It’s not very accurate.

For extreme accuracy, the atmospheric delay can be measured directly. To do this, a second transmission channel is incorporated.

The two satellite transmission channels are called L1 and L2. All receivers can use L1. L2 is encrypted. You have to have a secret key, available only to the U.S. government, to use the L2 channel. The two channels operate on somewhat different frequencies, and the atmosphere delays each channel differently.

And that’s all you need to know if you can receive both L1 and L2. You do not need to know in advance how the atmosphere delays each channel. The receiver can deduce the atmospheric delay from each satellite, and from that it can compute the position of the receiver to very high degree of accuracy.

When I first encountered this it was obvious to me there was not enough information to compute the atmospheric delay. So I asked a guy working on the project how this was supposed to work, and he stopped what he was doing and explained it to me. I still didn’t understand it, but I took my notes back to my cube and looked at it some more. It was an “oh shit” moment. “Of course, dummy.”

And that’s this week’s quiz question. How can a GPS receiver compute the atmospheric delay from the information given, using L1 and L2?

Post your answer as a comment below. I’m going to give this a few days and then post a hint.

Update and hint

I’ve had no activity on this Quiz Question all week. It’s time to provide a hint. Look at the problem again.

You have two radio signals originating from the same location at the same time and arriving at the receiver at different times. Because of ionospheric delay, you don’t know how long it took either signal to traverse the unknown distance to the transmitter. How can you use the information available to determine the distance to the transmitter. Here is the hint.

The satellite is moving. In the order of miles per second. Its distance from the receiver changes from one transmission to the next. How can you use multiple measurements to compensate for the ionospheric delay?

Update and solution

Time’s up. I need to post the solution to last week’s Quiz Question, because tomorrow’s question is going to be related to compensating for ionospheric delay. I’m not going to do the math. Instead, I’m going to pose the question in a different way that will make the solution obvious. It goes like this:

Forget satellites. There are two rail lines running parallel for miles over the horizon. You’re standing at a point along the rail lines waiting for two trains (call them A and B) to arrive. The two trains are going to start at the same time from the same location, and they are going to come at you at different speeds. You don’t know what the speeds are, but you know the speeds of A and B are different, and they are constant.

Trains A and B arrive. A arrives shortly before B. You note the time difference. You don’t have enough information to determine how far away the station (starting point) is. You move down the line a few miles, carefully noting how far you move.

Two more trains, also labeled A and B head your way from the station. Same as before. A arrives, then B arrives later. You record the time difference. Of course the difference is greater, because you are farther from the station, so the trains have had longer to diverge.

Now you ask yourself this question. “How far do I have to walk toward the direction the trains came from for the difference to be zero?” That’s the distance to the station.

In the case of GPS with channels L1 and L2 it’s the satellite that moves, and since the satellite is always telling you where it is, you know how much farther (or closer) it has moved from you between two measurements.

There is obviously more to it, so if anybody still has questions, post a comment and extend the dialogue.